#8702631

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#37644	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5341 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 6 generators on 32 points:

(10,13)(11,15)(18,24)(20,29)(25,26)(27,30);(1,2)(3,6)(4,7)(5,8)(9,10,12,13)(11,14,15,16)(17,18)(19,25)(20,22)(21,26)(23,24)(27,31)(28,30)(29,32);(1,3)(2,5)(4,8)(6,7)(9,10)(11,14)(12,13)(15,16)(17,19,23,21)(18,25,24,26)(20,30,29,27)(22,28,32,31);(3,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,20,23,29)(18,22,24,32)(19,27,21,30)(25,31,26,28);(1,2)(3,5)(4,7)(6,8)(9,11,12,15)(10,14,13,16)(17,21,23,19)(18,26,24,25)(20,30,29,27)(22,28,32,31);(1,3)(2,6)(4,8)(5,7)(17,22)(18,20)(19,28)(21,31)(23,32)(24,29)(25,30)(26,27)

Polycyclic presentation:

2^9|b^2=h,c^2=gh,d^2=gh,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8609827);